Math in the real world: Trees

The growth of trees is actually a fairly mathematical processthat at least involves fractal theory, graph theory, and topology. You can actually generate very realistic looking trees using a computer. See the video below for an example of simulated tree growth.

Here’s an idea. Take your kids outside and find some trees (even bushes or ferns will do in a pinch). Explore (and catalog) what rules different trees seem to follow as they branch. See if you can follow those same rules with pencil and paper to produce tree-like drawings. For bonus points, take some pictures of some younger trees, and use your rules to predict where the next branches will start, then follow up in a year to see if you were right.

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My dad is a hydrologist specializing in rivers and streams so I’ve been hiking up and down stream branches my whole life. He uses the branching patterns to, among other things, find “missing” tributaries. I used the same method when I introduced logarithms to my college math students. Here’s how it works:

Call the smallest streams or twigs, the ones at the very ends of the branches, Order 1. There are a lot of them! When two Order 1 branches meet, the bigger branch us called Order 2. When 2’s meet, you get Order 3, and so on. When a small meets a larger branch, though, you don’t increase the order (a big branch or river doesn’t change much when a littler one joins). Work your way down the entire tree (or watershed) until you reach the trunk (or river delta). You might be up to Order 4 or 5.

Now the cool part: count how many Order 1, 2,… there are and graph number vs order. On a normal plot, graph rapidly decays. Instead, plot log(number) vs Order: almost always a straight line! How does Nature know about exponentials? Wow!

In the case of rivers, you can extrapolate to smaller streams: suppose you had better maps that show even smaller creeks – how many Order 0, Order -1,…can you expect to find?

This works beautifully in class with a big tree branch (deciduous, that is. Conifers are different) you drag in. Students attach labels with Order to all the branches. Same relationships hold in landslides, air tubes in lungs, blood vessels,… In fact, anywhere where something must be distributed (air, blood) or collected (water, run-off) in the most efficient way possible.